Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, H

Percentage Accurate: 95.4% → 98.3%

Time: 5.2s

Alternatives: 11

Speedup: 1.0×

• 28.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - \frac{t}{y}\\ \mathbf{if}\;y \leq -2 \cdot 10^{-144}:\\ \;\;\;\;x - \frac{\frac{t\_1}{3}}{z}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t\_1}{3 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- y (/ t y))))
   (if (<= y -2e-144)
     (- x (/ (/ t_1 3.0) z))
     (if (<= y 8.6e-100)
       (fma (/ t z) (/ 0.3333333333333333 y) x)
       (- x (/ t_1 (* 3.0 z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y - (t / y);
	double tmp;
	if (y <= -2e-144) {
		tmp = x - ((t_1 / 3.0) / z);
	} else if (y <= 8.6e-100) {
		tmp = fma((t / z), (0.3333333333333333 / y), x);
	} else {
		tmp = x - (t_1 / (3.0 * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y - Float64(t / y))
	tmp = 0.0
	if (y <= -2e-144)
		tmp = Float64(x - Float64(Float64(t_1 / 3.0) / z));
	elseif (y <= 8.6e-100)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), x);
	else
		tmp = Float64(x - Float64(t_1 / Float64(3.0 * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2e-144], N[(x - N[(N[(t$95$1 / 3.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.6e-100], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(t$95$1 / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.9999999999999999e-144

   1. Initial program 98.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

      8. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      9. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 98.8

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 98.8

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 98.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{3 \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      3. associate-/r* N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

      5. lower-/.f64 98.8

         \[\leadsto x - \frac{\color{blue}{\frac{y - \frac{t}{y}}{3}}}{z} \]

   6. Applied rewrites 98.8%

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

   if -1.9999999999999999e-144 < y < 8.59999999999999997e-100

   1. Initial program 93.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z} + x \cdot y}{y}} \]
   4. Step-by-step derivation

      1. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z}}{y} + \frac{x \cdot y}{y}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{t}{z}}{y}} + \frac{x \cdot y}{y} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{t}{z \cdot y}} + \frac{x \cdot y}{y} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{\color{blue}{y \cdot z}} + \frac{x \cdot y}{y} \]

      5. associate-/l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x \cdot \frac{y}{y}} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{y}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y}{y}\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{y}{y}\right)\right)} \]

      9. *-inverses N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \color{blue}{-1} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{-1 \cdot x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot x \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} + 1 \cdot x} \]

      14. *-lft-identity N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x} \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} + x \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, x\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t}{y \cdot z}}, \frac{1}{3}, x\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{z \cdot y}}, \frac{1}{3}, x\right) \]

      19. lower-*.f64 93.5

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{z \cdot y}}, 0.3333333333333333, x\right) \]

   5. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(\frac{t}{z}, \color{blue}{\frac{0.3333333333333333}{y}}, x\right) \]

   if 8.59999999999999997e-100 < y

   1. Initial program 98.6%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

      8. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      9. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.8

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.8

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{-145} \lor \neg \left(y \leq 8.6 \cdot 10^{-100}\right):\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -6.1e-145) (not (<= y 8.6e-100)))
   (- x (/ (- y (/ t y)) (* 3.0 z)))
   (fma (/ t z) (/ 0.3333333333333333 y) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -6.1e-145) || !(y <= 8.6e-100)) {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	} else {
		tmp = fma((t / z), (0.3333333333333333 / y), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -6.1e-145) || !(y <= 8.6e-100))
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	else
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -6.1e-145], N[Not[LessEqual[y, 8.6e-100]], $MachinePrecision]], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.1e-145 or 8.59999999999999997e-100 < y

   1. Initial program 98.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

      8. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      9. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.3

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.3

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]

   if -6.1e-145 < y < 8.59999999999999997e-100

   1. Initial program 93.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z} + x \cdot y}{y}} \]
   4. Step-by-step derivation

      1. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z}}{y} + \frac{x \cdot y}{y}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{t}{z}}{y}} + \frac{x \cdot y}{y} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{t}{z \cdot y}} + \frac{x \cdot y}{y} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{\color{blue}{y \cdot z}} + \frac{x \cdot y}{y} \]

      5. associate-/l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x \cdot \frac{y}{y}} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{y}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y}{y}\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{y}{y}\right)\right)} \]

      9. *-inverses N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \color{blue}{-1} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{-1 \cdot x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot x \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} + 1 \cdot x} \]

      14. *-lft-identity N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x} \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} + x \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, x\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t}{y \cdot z}}, \frac{1}{3}, x\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{z \cdot y}}, \frac{1}{3}, x\right) \]

      19. lower-*.f64 93.5

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{z \cdot y}}, 0.3333333333333333, x\right) \]

   5. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(\frac{t}{z}, \color{blue}{\frac{0.3333333333333333}{y}}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{-145} \lor \neg \left(y \leq 8.6 \cdot 10^{-100}\right):\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(y \cdot z\right) \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* y z) 3.0))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((y * z) * 3.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + (t / ((y * z) * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((y * z) * 3.0));
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((y * z) * 3.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(y * z) * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((y * z) * 3.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(y * z), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.2%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}} \]

   2. *-commutative N/A

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \color{blue}{\left(z \cdot 3\right)}} \]

   4. associate-*r* N/A

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(y \cdot z\right) \cdot 3}} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(y \cdot z\right) \cdot 3}} \]

   6. lower-*.f64 97.2

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(y \cdot z\right)} \cdot 3} \]

4. Applied rewrites 97.2%

   \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(y \cdot z\right) \cdot 3}} \]
5. Add Preprocessing

Alternative 4: 92.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -80:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{0.3333333333333333 \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -80.0)
   (fma -0.3333333333333333 (/ y z) x)
   (if (<= y 8.2e+52)
     (fma (/ t z) (/ 0.3333333333333333 y) x)
     (- x (/ (* 0.3333333333333333 y) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -80.0) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else if (y <= 8.2e+52) {
		tmp = fma((t / z), (0.3333333333333333 / y), x);
	} else {
		tmp = x - ((0.3333333333333333 * y) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -80.0)
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	elseif (y <= 8.2e+52)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), x);
	else
		tmp = Float64(x - Float64(Float64(0.3333333333333333 * y) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -80.0], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 8.2e+52], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(N[(0.3333333333333333 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -80

   1. Initial program 98.3%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right)} \]

      2. *-lft-identity N/A

         \[\leadsto y \cdot \left(\color{blue}{1 \cdot \frac{x}{y}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      3. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      4. distribute-lft-neg-out N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      5. *-commutative N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{x}{y} \cdot -1}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      6. distribute-lft-neg-out N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot -1} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      7. mul-1-neg N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \cdot -1 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      8. distribute-lft-neg-out N/A

         \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)}\right) \]

      9. *-rgt-identity N/A

         \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot 1}\right)\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \color{blue}{-1}\right) \]

      12. distribute-rgt-in N/A

          \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      13. associate-*l* N/A

          \[\leadsto \color{blue}{\left(y \cdot -1\right) \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)} \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right)} \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \]

      15. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      17. mul-1-neg N/A

          \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      18. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)} \]

      19. +-commutative N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z} + -1 \cdot \frac{x}{y}\right)} \]

      20. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(\frac{1}{3} \cdot \frac{1}{z}\right) + \left(-1 \cdot y\right) \cdot \left(-1 \cdot \frac{x}{y}\right)} \]

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]

   if -80 < y < 8.1999999999999999e52

   1. Initial program 95.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z} + x \cdot y}{y}} \]
   4. Step-by-step derivation

      1. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z}}{y} + \frac{x \cdot y}{y}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{t}{z}}{y}} + \frac{x \cdot y}{y} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{t}{z \cdot y}} + \frac{x \cdot y}{y} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{\color{blue}{y \cdot z}} + \frac{x \cdot y}{y} \]

      5. associate-/l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x \cdot \frac{y}{y}} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{y}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y}{y}\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{y}{y}\right)\right)} \]

      9. *-inverses N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \color{blue}{-1} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{-1 \cdot x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot x \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} + 1 \cdot x} \]

      14. *-lft-identity N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x} \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} + x \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, x\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t}{y \cdot z}}, \frac{1}{3}, x\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{z \cdot y}}, \frac{1}{3}, x\right) \]

      19. lower-*.f64 88.8

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{z \cdot y}}, 0.3333333333333333, x\right) \]

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.2%

      \[\leadsto \mathsf{fma}\left(\frac{t}{z}, \color{blue}{\frac{0.3333333333333333}{y}}, x\right) \]

   if 8.1999999999999999e52 < y

   1. Initial program 99.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

      8. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      9. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.8

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.8

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{3 \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      3. associate-/r* N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

      5. lower-/.f64 99.8

         \[\leadsto x - \frac{\color{blue}{\frac{y - \frac{t}{y}}{3}}}{z} \]

   6. Applied rewrites 99.8%

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

   7. Taylor expanded in y around inf

      \[\leadsto x - \frac{\color{blue}{\frac{1}{3} \cdot y}}{z} \]
   8. Step-by-step derivation

      1. lower-*.f64 99.9

         \[\leadsto x - \frac{\color{blue}{0.3333333333333333 \cdot y}}{z} \]

   9. Applied rewrites 99.9%

      \[\leadsto x - \frac{\color{blue}{0.3333333333333333 \cdot y}}{z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -80:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{0.3333333333333333 \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -80.0)
   (fma -0.3333333333333333 (/ y z) x)
   (if (<= y 9.5e+54)
     (fma (/ t (* z y)) 0.3333333333333333 x)
     (- x (/ (* 0.3333333333333333 y) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -80.0) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else if (y <= 9.5e+54) {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	} else {
		tmp = x - ((0.3333333333333333 * y) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -80.0)
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	elseif (y <= 9.5e+54)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	else
		tmp = Float64(x - Float64(Float64(0.3333333333333333 * y) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -80.0], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 9.5e+54], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(x - N[(N[(0.3333333333333333 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -80

   1. Initial program 98.3%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right)} \]

      2. *-lft-identity N/A

         \[\leadsto y \cdot \left(\color{blue}{1 \cdot \frac{x}{y}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      3. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      4. distribute-lft-neg-out N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      5. *-commutative N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{x}{y} \cdot -1}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      6. distribute-lft-neg-out N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot -1} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      7. mul-1-neg N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \cdot -1 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      8. distribute-lft-neg-out N/A

         \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)}\right) \]

      9. *-rgt-identity N/A

         \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot 1}\right)\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \color{blue}{-1}\right) \]

      12. distribute-rgt-in N/A

          \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      13. associate-*l* N/A

          \[\leadsto \color{blue}{\left(y \cdot -1\right) \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)} \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right)} \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \]

      15. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      17. mul-1-neg N/A

          \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      18. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)} \]

      19. +-commutative N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z} + -1 \cdot \frac{x}{y}\right)} \]

      20. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(\frac{1}{3} \cdot \frac{1}{z}\right) + \left(-1 \cdot y\right) \cdot \left(-1 \cdot \frac{x}{y}\right)} \]

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]

   if -80 < y < 9.4999999999999999e54

   1. Initial program 95.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z} + x \cdot y}{y}} \]
   4. Step-by-step derivation

      1. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z}}{y} + \frac{x \cdot y}{y}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{t}{z}}{y}} + \frac{x \cdot y}{y} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{t}{z \cdot y}} + \frac{x \cdot y}{y} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{\color{blue}{y \cdot z}} + \frac{x \cdot y}{y} \]

      5. associate-/l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x \cdot \frac{y}{y}} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{y}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y}{y}\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{y}{y}\right)\right)} \]

      9. *-inverses N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \color{blue}{-1} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{-1 \cdot x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot x \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} + 1 \cdot x} \]

      14. *-lft-identity N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x} \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} + x \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, x\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t}{y \cdot z}}, \frac{1}{3}, x\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{z \cdot y}}, \frac{1}{3}, x\right) \]

      19. lower-*.f64 88.8

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{z \cdot y}}, 0.3333333333333333, x\right) \]

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)} \]

   if 9.4999999999999999e54 < y

   1. Initial program 99.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

      8. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      9. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.8

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.8

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{3 \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      3. associate-/r* N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

      5. lower-/.f64 99.8

         \[\leadsto x - \frac{\color{blue}{\frac{y - \frac{t}{y}}{3}}}{z} \]

   6. Applied rewrites 99.8%

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

   7. Taylor expanded in y around inf

      \[\leadsto x - \frac{\color{blue}{\frac{1}{3} \cdot y}}{z} \]
   8. Step-by-step derivation

      1. lower-*.f64 99.9

         \[\leadsto x - \frac{\color{blue}{0.3333333333333333 \cdot y}}{z} \]

   9. Applied rewrites 99.9%

      \[\leadsto x - \frac{\color{blue}{0.3333333333333333 \cdot y}}{z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 88.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -80:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{z \cdot y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{0.3333333333333333 \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -80.0)
   (fma -0.3333333333333333 (/ y z) x)
   (if (<= y 9.5e+54)
     (fma t (/ 0.3333333333333333 (* z y)) x)
     (- x (/ (* 0.3333333333333333 y) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -80.0) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else if (y <= 9.5e+54) {
		tmp = fma(t, (0.3333333333333333 / (z * y)), x);
	} else {
		tmp = x - ((0.3333333333333333 * y) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -80.0)
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	elseif (y <= 9.5e+54)
		tmp = fma(t, Float64(0.3333333333333333 / Float64(z * y)), x);
	else
		tmp = Float64(x - Float64(Float64(0.3333333333333333 * y) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -80.0], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 9.5e+54], N[(t * N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(N[(0.3333333333333333 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -80

   1. Initial program 98.3%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right)} \]

      2. *-lft-identity N/A

         \[\leadsto y \cdot \left(\color{blue}{1 \cdot \frac{x}{y}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      3. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      4. distribute-lft-neg-out N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      5. *-commutative N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{x}{y} \cdot -1}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      6. distribute-lft-neg-out N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot -1} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      7. mul-1-neg N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \cdot -1 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      8. distribute-lft-neg-out N/A

         \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)}\right) \]

      9. *-rgt-identity N/A

         \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot 1}\right)\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \color{blue}{-1}\right) \]

      12. distribute-rgt-in N/A

          \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      13. associate-*l* N/A

          \[\leadsto \color{blue}{\left(y \cdot -1\right) \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)} \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right)} \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \]

      15. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      17. mul-1-neg N/A

          \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      18. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)} \]

      19. +-commutative N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z} + -1 \cdot \frac{x}{y}\right)} \]

      20. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(\frac{1}{3} \cdot \frac{1}{z}\right) + \left(-1 \cdot y\right) \cdot \left(-1 \cdot \frac{x}{y}\right)} \]

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]

   if -80 < y < 9.4999999999999999e54

   1. Initial program 95.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z} + x \cdot y}{y}} \]
   4. Step-by-step derivation

      1. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z}}{y} + \frac{x \cdot y}{y}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{t}{z}}{y}} + \frac{x \cdot y}{y} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{t}{z \cdot y}} + \frac{x \cdot y}{y} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{\color{blue}{y \cdot z}} + \frac{x \cdot y}{y} \]

      5. associate-/l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x \cdot \frac{y}{y}} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{y}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y}{y}\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{y}{y}\right)\right)} \]

      9. *-inverses N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \color{blue}{-1} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{-1 \cdot x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot x \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} + 1 \cdot x} \]

      14. *-lft-identity N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x} \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} + x \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, x\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t}{y \cdot z}}, \frac{1}{3}, x\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{z \cdot y}}, \frac{1}{3}, x\right) \]

      19. lower-*.f64 88.8

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{z \cdot y}}, 0.3333333333333333, x\right) \]

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 88.2%

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{0.3333333333333333}{z \cdot y}}, x\right) \]

   if 9.4999999999999999e54 < y

   1. Initial program 99.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

      8. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      9. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.8

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.8

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{3 \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      3. associate-/r* N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

      5. lower-/.f64 99.8

         \[\leadsto x - \frac{\color{blue}{\frac{y - \frac{t}{y}}{3}}}{z} \]

   6. Applied rewrites 99.8%

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

   7. Taylor expanded in y around inf

      \[\leadsto x - \frac{\color{blue}{\frac{1}{3} \cdot y}}{z} \]
   8. Step-by-step derivation

      1. lower-*.f64 99.9

         \[\leadsto x - \frac{\color{blue}{0.3333333333333333 \cdot y}}{z} \]

   9. Applied rewrites 99.9%

      \[\leadsto x - \frac{\color{blue}{0.3333333333333333 \cdot y}}{z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{t}{z \cdot y} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;x - \frac{0.3333333333333333 \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.05e-11)
   (fma -0.3333333333333333 (/ y z) x)
   (if (<= y 2.3e-95)
     (* (/ t (* z y)) 0.3333333333333333)
     (- x (/ (* 0.3333333333333333 y) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.05e-11) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else if (y <= 2.3e-95) {
		tmp = (t / (z * y)) * 0.3333333333333333;
	} else {
		tmp = x - ((0.3333333333333333 * y) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.05e-11)
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	elseif (y <= 2.3e-95)
		tmp = Float64(Float64(t / Float64(z * y)) * 0.3333333333333333);
	else
		tmp = Float64(x - Float64(Float64(0.3333333333333333 * y) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.05e-11], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 2.3e-95], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(x - N[(N[(0.3333333333333333 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.0499999999999999e-11

   1. Initial program 98.4%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right)} \]

      2. *-lft-identity N/A

         \[\leadsto y \cdot \left(\color{blue}{1 \cdot \frac{x}{y}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      3. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      4. distribute-lft-neg-out N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      5. *-commutative N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{x}{y} \cdot -1}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      6. distribute-lft-neg-out N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot -1} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      7. mul-1-neg N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \cdot -1 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      8. distribute-lft-neg-out N/A

         \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)}\right) \]

      9. *-rgt-identity N/A

         \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot 1}\right)\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \color{blue}{-1}\right) \]

      12. distribute-rgt-in N/A

          \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      13. associate-*l* N/A

          \[\leadsto \color{blue}{\left(y \cdot -1\right) \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)} \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right)} \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \]

      15. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      17. mul-1-neg N/A

          \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      18. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)} \]

      19. +-commutative N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z} + -1 \cdot \frac{x}{y}\right)} \]

      20. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(\frac{1}{3} \cdot \frac{1}{z}\right) + \left(-1 \cdot y\right) \cdot \left(-1 \cdot \frac{x}{y}\right)} \]

   5. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]

   if -1.0499999999999999e-11 < y < 2.29999999999999999e-95

   1. Initial program 95.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t}{y \cdot z}} \cdot \frac{1}{3} \]

      4. *-commutative N/A

         \[\leadsto \frac{t}{\color{blue}{z \cdot y}} \cdot \frac{1}{3} \]

      5. lower-*.f64 69.0

         \[\leadsto \frac{t}{\color{blue}{z \cdot y}} \cdot 0.3333333333333333 \]

   5. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\frac{t}{z \cdot y} \cdot 0.3333333333333333} \]

   if 2.29999999999999999e-95 < y

   1. Initial program 98.6%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

      8. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      9. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.8

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.8

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{3 \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      3. associate-/r* N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

      5. lower-/.f64 99.8

         \[\leadsto x - \frac{\color{blue}{\frac{y - \frac{t}{y}}{3}}}{z} \]

   6. Applied rewrites 99.8%

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

   7. Taylor expanded in y around inf

      \[\leadsto x - \frac{\color{blue}{\frac{1}{3} \cdot y}}{z} \]
   8. Step-by-step derivation

      1. lower-*.f64 89.3

         \[\leadsto x - \frac{\color{blue}{0.3333333333333333 \cdot y}}{z} \]

   9. Applied rewrites 89.3%

      \[\leadsto x - \frac{\color{blue}{0.3333333333333333 \cdot y}}{z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot t}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{0.3333333333333333 \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.05e-11)
   (fma -0.3333333333333333 (/ y z) x)
   (if (<= y 2.3e-95)
     (/ (* 0.3333333333333333 t) (* z y))
     (- x (/ (* 0.3333333333333333 y) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.05e-11) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else if (y <= 2.3e-95) {
		tmp = (0.3333333333333333 * t) / (z * y);
	} else {
		tmp = x - ((0.3333333333333333 * y) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.05e-11)
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	elseif (y <= 2.3e-95)
		tmp = Float64(Float64(0.3333333333333333 * t) / Float64(z * y));
	else
		tmp = Float64(x - Float64(Float64(0.3333333333333333 * y) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.05e-11], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 2.3e-95], N[(N[(0.3333333333333333 * t), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(0.3333333333333333 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.0499999999999999e-11

   1. Initial program 98.4%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right)} \]

      2. *-lft-identity N/A

         \[\leadsto y \cdot \left(\color{blue}{1 \cdot \frac{x}{y}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      3. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      4. distribute-lft-neg-out N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      5. *-commutative N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{x}{y} \cdot -1}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      6. distribute-lft-neg-out N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot -1} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      7. mul-1-neg N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \cdot -1 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

      8. distribute-lft-neg-out N/A

         \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)}\right) \]

      9. *-rgt-identity N/A

         \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot 1}\right)\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \color{blue}{-1}\right) \]

      12. distribute-rgt-in N/A

          \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      13. associate-*l* N/A

          \[\leadsto \color{blue}{\left(y \cdot -1\right) \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)} \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right)} \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \]

      15. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      17. mul-1-neg N/A

          \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

      18. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)} \]

      19. +-commutative N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z} + -1 \cdot \frac{x}{y}\right)} \]

      20. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(\frac{1}{3} \cdot \frac{1}{z}\right) + \left(-1 \cdot y\right) \cdot \left(-1 \cdot \frac{x}{y}\right)} \]

   5. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]

   if -1.0499999999999999e-11 < y < 2.29999999999999999e-95

   1. Initial program 95.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z} + x \cdot y}{y}} \]
   4. Step-by-step derivation

      1. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z}}{y} + \frac{x \cdot y}{y}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{t}{z}}{y}} + \frac{x \cdot y}{y} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{t}{z \cdot y}} + \frac{x \cdot y}{y} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{\color{blue}{y \cdot z}} + \frac{x \cdot y}{y} \]

      5. associate-/l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x \cdot \frac{y}{y}} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{y}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y}{y}\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{y}{y}\right)\right)} \]

      9. *-inverses N/A

         \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - x \cdot \color{blue}{-1} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{-1 \cdot x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot x \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} + 1 \cdot x} \]

      14. *-lft-identity N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{x} \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} + x \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, x\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t}{y \cdot z}}, \frac{1}{3}, x\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{z \cdot y}}, \frac{1}{3}, x\right) \]

      19. lower-*.f64 92.2

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{z \cdot y}}, 0.3333333333333333, x\right) \]

   5. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 91.4%

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{0.3333333333333333}{z \cdot y}}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{t}{y \cdot z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.7%

       \[\leadsto \frac{\frac{t}{y}}{z} \cdot \color{blue}{0.3333333333333333} \]
   11. Step-by-step derivation

   12. Applied rewrites 69.0%

       \[\leadsto \frac{0.3333333333333333 \cdot t}{z \cdot \color{blue}{y}} \]

   if 2.29999999999999999e-95 < y

   1. Initial program 98.6%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

      8. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      9. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.8

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.8

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{3 \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      3. associate-/r* N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

      5. lower-/.f64 99.8

         \[\leadsto x - \frac{\color{blue}{\frac{y - \frac{t}{y}}{3}}}{z} \]

   6. Applied rewrites 99.8%

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

   7. Taylor expanded in y around inf

      \[\leadsto x - \frac{\color{blue}{\frac{1}{3} \cdot y}}{z} \]
   8. Step-by-step derivation

      1. lower-*.f64 89.3

         \[\leadsto x - \frac{\color{blue}{0.3333333333333333 \cdot y}}{z} \]

   9. Applied rewrites 89.3%

      \[\leadsto x - \frac{\color{blue}{0.3333333333333333 \cdot y}}{z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma -0.3333333333333333 (/ y z) x))

C

double code(double x, double y, double z, double t) {
	return fma(-0.3333333333333333, (y / z), x);
}

Julia

function code(x, y, z, t)
	return fma(-0.3333333333333333, Float64(y / z), x)
end

Wolfram

code[x_, y_, z_, t_] := N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 97.2%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

      \[\leadsto y \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right)} \]

   2. *-lft-identity N/A

      \[\leadsto y \cdot \left(\color{blue}{1 \cdot \frac{x}{y}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

   3. metadata-eval N/A

      \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \frac{x}{y} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

   4. distribute-lft-neg-out N/A

      \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{x}{y}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

   5. *-commutative N/A

      \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{x}{y} \cdot -1}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

   6. distribute-lft-neg-out N/A

      \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot -1} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

   7. mul-1-neg N/A

      \[\leadsto y \cdot \left(\color{blue}{\left(-1 \cdot \frac{x}{y}\right)} \cdot -1 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{z}\right) \]

   8. distribute-lft-neg-out N/A

      \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{z}\right)\right)}\right) \]

   9. *-rgt-identity N/A

      \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot 1}\right)\right)\right) \]

   10. distribute-rgt-neg-in N/A

       \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto y \cdot \left(\left(-1 \cdot \frac{x}{y}\right) \cdot -1 + \left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot \color{blue}{-1}\right) \]

   12. distribute-rgt-in N/A

       \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

   13. associate-*l* N/A

       \[\leadsto \color{blue}{\left(y \cdot -1\right) \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)} \]

   14. *-commutative N/A

       \[\leadsto \color{blue}{\left(-1 \cdot y\right)} \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \]

   15. mul-1-neg N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right) \]

   16. distribute-lft-neg-in N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

   17. mul-1-neg N/A

       \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)\right)} \]

   18. associate-*r* N/A

       \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{3} \cdot \frac{1}{z}\right)} \]

   19. +-commutative N/A

       \[\leadsto \left(-1 \cdot y\right) \cdot \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{z} + -1 \cdot \frac{x}{y}\right)} \]

   20. distribute-lft-in N/A

       \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(\frac{1}{3} \cdot \frac{1}{z}\right) + \left(-1 \cdot y\right) \cdot \left(-1 \cdot \frac{x}{y}\right)} \]

5. Applied rewrites 66.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]
6. Add Preprocessing

Alternative 10: 35.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{y}{z} \cdot -0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (/ y z) -0.3333333333333333))

C

double code(double x, double y, double z, double t) {
	return (y / z) * -0.3333333333333333;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y / z) * (-0.3333333333333333d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (y / z) * -0.3333333333333333;
}

Python

def code(x, y, z, t):
	return (y / z) * -0.3333333333333333

Julia

function code(x, y, z, t)
	return Float64(Float64(y / z) * -0.3333333333333333)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y / z) * -0.3333333333333333;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y / z), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 97.2%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. associate-+l- N/A

      \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

   5. lift-/.f64 N/A

      \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

   8. *-commutative N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

   9. associate-/r* N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

   10. sub-div N/A

       \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

   11. lower-/.f64 N/A

       \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

   12. lower--.f64 N/A

       \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

   13. lower-/.f64 93.6

       \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

   14. lift-*.f64 N/A

       \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   15. *-commutative N/A

       \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   16. lower-*.f64 93.6

       \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

4. Applied rewrites 93.6%

   \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{3 \cdot z}} \]

   2. lift-*.f64 N/A

      \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   3. associate-/r* N/A

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

   4. lower-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

   5. lower-/.f64 93.6

      \[\leadsto x - \frac{\color{blue}{\frac{y - \frac{t}{y}}{3}}}{z} \]

6. Applied rewrites 93.6%

   \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

7. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{-1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{-1}{3}} \]

   3. lower-/.f64 40.0

      \[\leadsto \color{blue}{\frac{y}{z}} \cdot -0.3333333333333333 \]

9. Applied rewrites 40.0%

   \[\leadsto \color{blue}{\frac{y}{z} \cdot -0.3333333333333333} \]

10. Final simplification 40.0%

    \[\leadsto \frac{y}{z} \cdot -0.3333333333333333 \]
11. Add Preprocessing

Alternative 11: 35.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ y \cdot \frac{-0.3333333333333333}{z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* y (/ -0.3333333333333333 z)))

C

double code(double x, double y, double z, double t) {
	return y * (-0.3333333333333333 / z);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y * ((-0.3333333333333333d0) / z)
end function

Java

public static double code(double x, double y, double z, double t) {
	return y * (-0.3333333333333333 / z);
}

Python

def code(x, y, z, t):
	return y * (-0.3333333333333333 / z)

Julia

function code(x, y, z, t)
	return Float64(y * Float64(-0.3333333333333333 / z))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = y * (-0.3333333333333333 / z);
end

Wolfram

code[x_, y_, z_, t_] := N[(y * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.2%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. associate-+l- N/A

      \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

   5. lift-/.f64 N/A

      \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

   8. *-commutative N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

   9. associate-/r* N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

   10. sub-div N/A

       \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

   11. lower-/.f64 N/A

       \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

   12. lower--.f64 N/A

       \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

   13. lower-/.f64 93.6

       \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

   14. lift-*.f64 N/A

       \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   15. *-commutative N/A

       \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   16. lower-*.f64 93.6

       \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

4. Applied rewrites 93.6%

   \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{3 \cdot z}} \]

   2. lift-*.f64 N/A

      \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   3. associate-/r* N/A

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

   4. lower-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

   5. lower-/.f64 93.6

      \[\leadsto x - \frac{\color{blue}{\frac{y - \frac{t}{y}}{3}}}{z} \]

6. Applied rewrites 93.6%

   \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

7. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{-1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{-1}{3}} \]

   3. lower-/.f64 40.0

      \[\leadsto \color{blue}{\frac{y}{z}} \cdot -0.3333333333333333 \]

9. Applied rewrites 40.0%

   \[\leadsto \color{blue}{\frac{y}{z} \cdot -0.3333333333333333} \]
10. Step-by-step derivation

11. Applied rewrites 40.0%

    \[\leadsto y \cdot \color{blue}{\frac{-0.3333333333333333}{z}} \]

12. Final simplification 40.0%

    \[\leadsto y \cdot \frac{-0.3333333333333333}{z} \]
13. Add Preprocessing

Developer Target 1: 96.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y)))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + ((t / (z * 3.0d0)) / y)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y)

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(Float64(t / Float64(z * 3.0)) / y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(z * 3.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025017 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (- x (/ y (* z 3))) (/ (/ t (* z 3)) y)))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))